sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
.
,
,
is a positive integer.
of positive integers 
such that 
.
denotes component-wise function multiplication)
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
.
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
Y by Adventure10, Mango247
Let 
be a triangle inscribed in circle 
with center 
; let 
be its 
-mixtilinear incircle, 
be its 
-mixtilinear incircle, 
be its 
-mixtilinear incircle, and 
be the radical center of 
. Let 
, 
, 
be the points at which , , are tangent to . Prove that 
, 
, 
and 
are concurrent.
Proposed by Robin Park
be a triangle inscribed in circle 
with center 
; let 
be its 
-mixtilinear incircle, 
be its 
-mixtilinear incircle, 
be its 
-mixtilinear incircle, and 
be the radical center of 
. Let 
, 
, 
be the points at which , , are tangent to . Prove that 
, 
, 
and 
are concurrent.Proposed by Robin Park
be the incircle of 
and 
thus by Monge and d'Alembert theorem, 
at the exsimilicenter of 
and similarly 
i.e. 
As for the radical center 
it is the midpoint 
between 
of the intouch and excentral triangle, lying on 
